Let $n$ be a positive integer.

We say $\lambda (\in \mathbb{N}^n)$ is a partition of $n$ iff $\lambda_1\geq \cdots \geq \lambda_n \geq 0$ and $\lambda_1+\cdots +\lambda_k=n$.

Is there a standard notation to designate the set of all partitions of $n$?

  • $\begingroup$ Well, actually there is no typo here. Someone edited it to exclude the case $\lambda_k =0$ but I'm inclusing this case too. $\endgroup$ – Rubertos May 17 '17 at 5:43
  • $\begingroup$ we don't count 0's in partitions $\endgroup$ – JonMark Perry May 17 '17 at 5:44
  • $\begingroup$ Yes that's the definition for partition of sets, but I am following this note : hep.caltech.edu/~fcp/math/groupTheory/young.pdf $\endgroup$ – Rubertos May 17 '17 at 5:46
  • $\begingroup$ Anyway, if we exclude the case $\lambda_k\neq 0$, is there a standard notation for this? Would it be $\Pi_n$? $\endgroup$ – Rubertos May 17 '17 at 5:46
  • $\begingroup$ Here there is $P(n)$, but I suspect there is no standard notation. $\endgroup$ – Jean-Claude Arbaut May 17 '17 at 6:04

We write:

$$\mathbf\lambda\vdash n$$

to indicate
$$\mathbf\lambda=\{\lambda_1,\dots,\lambda_k\}, \;\;\lambda_i\gt0, \lambda_i\in\mathbb{N} $$ and $$\sum_\limits{i=1}^k \lambda_i=n$$

You have $k=n$ and $\lambda_i\ge 0$, but this shouldn't matter.

Capital $\lambda$ is $\Lambda$, so you could use this.


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